Saturday, June 23, 2012

Remarkable Relationships

I have long commented on the linear (1-dimensional) nature of Conventional Mathematics from a qualitative perspective. This implies a merely uni-polar approach i.e. where objective is clearly separated from subjective, quantitative from qualitative etc.

Put another way it implies that mathematical meaning is merely posited (in a real conscious rational manner).

Thus though negative and imaginary operations are certainly possible within Conventional Mathematics in quantitative terms, these are always strictly interpreted in a linear (1-dimensional) manner from a qualitative perspective.

Furthermore, as I have repeatedly pointed out, the conventional understanding of both the Prime Number Theorem and the Riemann Hypothesis takes place within this limited linear perspective.

As we know the circular roots of unity (apart from the 1st and 2nd dimensions) entail complex numbers with positive and negative values. However though these roots inherently entail a dynamic relationship as between linear and circular notions, in conventional mathematical terms they are interpreted in a merely reduced linear manner (i.e. where the qualitative aspect is directly reduced in quantitative terms).

However we can invert this whole approach whereby now mathematical operations are considered quantitatively in a real positive manner, with qualitative interpretation taking place in a multi-dimensional fashion.

What this entails is that we simply interpret with respect to quantitative operations both negative and imaginary values in a positive real fashion.

For example the 3 roots of 1 are 1, -.5 +.866i, and -.5 - .866i (correct to 3 decimal places).

Now interpreted in a real positive quantitative manner these are 1, .5 + .866 and .5 + .866 i.e. 1, 1.366 and 1.366.

Now if we obtain the mean average of these 3 roots, i.e. (1 + 1.366 + 1.366)/3 = 1.244.

This is already a reasonable approximation to the value of 4/π = 1.27323954...
and this approximation quickly increases as the number of roots of 1 also increases.

So confining ourselves to prime numbered roots, the mean average of the 17 roots of 1(in this real positive manner) = 1.2723335... which is already very close to 4/π =

Thus as we increase the number of prime numbered roots of 1, the mean average value approximates very quickly to 4/π.

Now the significance of 4/π is that it serves as a perfect archetype of linear to circular meaning representing both the length of the perimeter of the square to its inscribed circular circumference and equally the area of the same square to the area of its inscribed circle.

Now if we take the same n roots of 1 and then multiply them (in this real positive manner) before then raising to the power of 1/n once again the result will approximate to 4/π.

For example the product of the 3 roots of 1 i.e. 1 * 1.366 * 1.366 = 1.865956 and when we raise this to 1/3 we obtain 1.23112... which is already a reasonable approximation to 4/π = 1.27323954...

And this approximation to 4/π again improves (though more slowly than in the previous case) as we increase the number of roots of 1.

So again using 17 roots to illustrate, when we multiply these together and then raise to 1/17 we obtain 1.2657221... (which is much closer to 4/π).

Thus there is a remarkable link quantitatively here as between addition and multiplication on the one hand and multiplication and exponentiation on the other.

Therefore with respect to the n roots of a given prime number (taken in a reduced real positive manner),

(1st + 2nd + 3rd ... + nth)*(1/n) → (1st * 2nd * 3rd ... * nth)^(1/n) → 4/π with the approximation improving as n gets larger. So in the limit as n → ∞, equality as between the three expressions is achieved (in a relative manner).

In the past I have continually expressed the viewpoint that that nature of prime number behaviour cannot be properly understood without the incorporation of both linear (analytic) and circular (holistic) modes of qualitative interpretation.

Now here in a reverse quantitative manner we can see the same perfect relationship as between linear and circular notions with respect to number behaviour.

So once again this clearly demonstrates that it is not enough for example to attempt to define linear and circular notions in a merely quantitative manner (as in Type 1 Mathematics).

Equally they must be defined in a qualitative manner (Type 2 Mathematics).

And then comprehensive understanding can only materialise through the synchronised interaction of both quantitative and qualitative meaning (Type 3 Mathematics).

Thursday, June 21, 2012

Prime Number Theorem: Alternative Formulation

In a short addendum (on the "The Euler Identity" blog), a simple approximation for log n (with n positive)was provided i.e.

log n → (n^x - 1)/x as x → 0.

The Prime Number Theorem (providing the general distribution of the primes among the natural numbers) in turn can be simply expressed by the expression n/log n (as n → ∞).

Therefore substituting our approximation for log n, the Prime Number Theorem could be expressed as nx/(n^x - 1) as n → ∞ and x → 0!

It is often expressed that the secret of prime number behaviour lies in the relationship as between addition and multiplication!

It could equally be said that this secret lies in the relationship as between multiplication and exponentiation (which is simply demonstrated by this expression).

Also we have the combination of two extremes whereby the relative approximation is continually improved through making one variable (n) progressively larger, while the other variable (x) is made progressively smaller.

This likewise captures the essence of prime number behaviour which represents an extreme as between linear (quantitative) and circular (qualitative) aspects, whereby the relative independence of each prime number (as discrete) is made compatible with the overall interdependence of prime numbers (as continuous).

We also had provided a complementary Prime Number Theorem where the average absolute value of prime numbered roots of 1 approximates to 2/π = i/log i.

Once again i/log i can be approximated as ix/(i^x - 1) as x → 0.

So 2/π → ix/(i^x - 1) as x → 0.